Answer to Question #213093 in Real Analysis for Nikhil

Question

-2 is the limit point of the interval ]-3,2[.

True or false with full explanation

Answer 1

A limit point is a point for which every neighbourhood contains at least one point belonging to a given set.

Given the closed interval [-3,2], the point -2 is a limit point for the interval, since we can find a neighbourhood of -2 which completely lies in the interval.

Considering the above given closed interval, A neighbourhood of -2 is the open interval (-2.5,1). We can find a $\epsilon>0$ (a small number) such that

$$-2 \in (-2.5, 1) \subset (-2.5- \epsilon, 1+ \epsilon) \subset [-3,2]$$

Hence,

-2 is the limit point of the interval [-3,2].